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Unformatted text preview: Belle’s score.
(4) (c) Let X denote the largest number shown on the four dice.
4 (i) x
Show that for P(X ≤ x) = , for x = 1, 2,... 6
6 (ii) Copy and complete the following probability distribution table.
x
P(X = x) (iii) 1 2 1
15
1296 1296 3 4 5 6 671
1296 Calculate E(X).
(7)
(Total 13 marks) 298 438. The function f is defined by
f ( x) = x2 – x +1
x2 + x +1 (i) Find an expression for f′(x), simplifying your answer. (ii) (a) The tangents to the curve of f(x) at points A and B are parallel to the xaxis. Find
the coordinates of A and of B.
(5) (i) Sketch the graph of y = f′(x). (ii) (b) Find the xcoordinates of the three points of inflexion on the graph of f.
(5) (c) Find the range of
(i) f; (ii) the composite function f °f.
(5)
(Total 15 marks) 439. The random variable X is Poisson distributed with mean µ and satisfies P(X = 3) = P(X = 0) +
P(X = 1).
(a) Find the value of µ, correct to four decimal places.
(3) (b) For this value of µ evaluate P(2 ≤ X ≤ 4).
(3)
(Total 6 marks) 299 440. The weights of male nurses in a hospital are known to be normally distributed with mean
µ = 72 kg and standard deviation σ = 7.5kg. The hospital has a lift (elevator) with a maximum
recommended load of 450 kg. Six male nurses enter the lift. Calculate the probability p that
their combined weight exceeds the maximum recommended load.
(Total 5 marks) 441. Six coins are tossed simultaneously 320 times, with the following results.
0 tail 5 times 1 tail 40 times 2 tails 86 times 3 tails 89 times 4 tails 67 times 5 tails 29 times 6 tails 4 times At the 5% level of significance, test the hypothesis that all the coins are fair.
(Total 9 marks) 442. Let A, B and C be subsets of a given universal set.
(a) Use a Venn diagram to show that (A ∩ B) ∪ C =(A ∪ C) ∩ (B ∪ C).
(2) (b) Hence, and by using De Morgan’s laws, show that
(A′ ∩ B) ∪ C′ = (A ∩ C)′ ∩ (B′ ∩ C)′.
(3)
(Total 5 marks) 300 443. Let R be a relation on
x, y ∈ .
(a) such that for m ∈ + Prove that R is an equivalence relation on , x R y if and only if m divides x – y, where .
(4) (b) Prove that this equivalence relation partitions into m distinct classes.
(4) (c) Let m be the set of all the equivalence classes found in part (b). Define a suitable binary
operation +m on m and prove that ( m, + m) is an additive Abelian group.
(5) (d) Let (K,◊) be a cyclic group of order m. Prove that (K, ◊) is isomorphic to m. (4)
(Total 17 marks) 444. Let (G, °) be a group with subgroups (H, °) and (K, °). Prove that (H ∪ K, °) is a subgroup of
(G, °) if and only if one of the sets H and K is contained in the other.
(Total 8 marks) 445. (a) Use the Euclidean algorithm to find the greatest common divisor of 568 and 208.
(3) (b) Hence or otherwise, find two integers m and n such that 568m – 208n = 8.
(4)
(Total 7 marks) 301 446. (a) Define the isomorphism of two graphs G and H.
(3) (b) Determine whether the two graphs below are isomorphic. Give a reason for your ans...
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 Fall '13
 Apple
 The Land

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